Wall mode dynamics and transition to chaos in magnetoconvection with a vertical magnetic field

TL;DR

This study uses long 3D simulations to explore how wall mode magnetoconvection transitions to chaos under different magnetic field strengths, revealing bifurcations and coexistence of flow structures in low Prandtl number fluids.

Contribution

It provides detailed mapping of bifurcations and chaotic transitions in wall mode magnetoconvection at high Hartmann numbers, which was previously unexplored.

Findings

Wall modes undergo symmetry-breaking bifurcations.

Coexistence of wall modes and large-scale rolls observed.

Chaotic dynamics arise from wall mode chaos and cellular structures.

Abstract

Quasistatic magnetoconvection of a low Prandtl number fluid ($\textrm{Pr}=0.025)$ with a vertical magnetic field is considered in a unit aspect ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number $\textrm{Ha}$, the onset of convection is known to result from a sidewall instability giving rise to the wall mode regime. Here, we carry out 3D direct numerical simulations of unprecedented length to map out the parameter space at $\textrm{Ha} = 200, 500, 1000$, varying the Rayleigh number ($\textrm{Ra}$) between $6\times10^5 \lesssim \textrm{Ra} \lesssim 5\times 10^8$. We track the development of stable equilibria produced by this primary instability, identify bifurcations leading to limit cycles, and eventually to chaotic dynamics. At {$\textrm{Ha}=200$}, the steady wall mode solution undergoes a symmetry-breaking bifurcation producing a state featuring a coexistence between wall modes and a large-scale roll in the centre of the domain which persists to higher $\textrm{Ra}$. However, under a stronger magnetic field at $\textrm{Ha}=1000$, the steady wall mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in $\textrm{Ra}$, the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of $\textrm{Ra}$, and chaotic dynamics primarily arise due to a mixture of chaotic wall mode dynamics and arrays of cellular structures.